Counter-example in 3D and homogenization of geometric motions in 2D (long version)

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1 Counter-example in 3D and homogenization of geometric motions in D long version L.A. Caffarelli, R. Monneau July 6, 01 Abstract In this paper we give a counter-example to the homogenization of the forced mean curvature motion in a periodic setting in dimension N 3 when the forcing is positive. We also prove a general homogenization result for geometric motions in dimension N = under the assumption that there exists a constant δ > 0 such that every straight line moving with a normal velocity equal to δ is a subsolution for the motion. We also present a generalization in dimension, where we allow sign changing normal velocity and still construct bounded correctors, when there exists a subsolution with compact support expanding in all directions. AMS Classification: 35B7, 35K55, 35J0. Keywords: homogenization, mean curvature motion, geometric motion, propagation of fronts, heterogeneous media, periodic media, viscosity solutions. Contents 1 Introduction 1.1 Setting of the problem Main results Brief review of the literature Organization of the paper Strategy of the proofs 8.1 The counter-example in 3D Theorem The cell problem in D Theorem Homogenization Theorem Department of Mathematics, Institute for Computational Engineering and Sciences, University of Texas at Austin, 1 University Station, C100, Austin, Texas , USA Université Paris Est, CERMICS, Ecole des Ponts ParisTech, 6 et 8 avenue Blaise Pascal, Cité Descartes, Champs-sur-Marne, Marne-la-Vallée Cedex, FRANCE 1

2 3 Properties of viscosity solutions Viscosity solutions The technical assumption A An example More properties on viscosity solutions Subdifferentials Counter-example in dimension N Preliminaries in any dimension 3 6 Flatness of E t for N = 3 7 Existence of a corrector: proof of Theorem Conditional homogenization in any dimension: proof of Theorem Homogenization in D: proof of Theorem The cell problem in D with sign changing normal velocity New preliminary results Revisiting Section Revisiting Section Revisiting Section 7 and proof of Theorem Examples and applications The case where c is not positive The G-equation with large divergence vector field Appendix Barriers from the initial data Technical lemmata used in the proof of Theorem Proof of the comparison principle Theorem Introduction 1.1 Setting of the problem In this paper we are interested in solutions u x, t for > 0 of geometric equations that can be written as u t = F D u, Du, 1 x on R N 0, u = u 0 on R N {0} for suitable F which are in particular periodic in the variable 1 x. Under certain assumptions we can show the homogenization as 0, i.e. that u converges to a function u 0

3 solution of an equation 1. u 0 t = F Du 0 on R N 0, + u 0 = u 0 on R N {0}. Our starting point is the study of the mean curvature motion forced by a given periodic verlocity c, i.e. { u t = tr D u I Du Du } + c 1 x Du on R N 0, u = u 0 on R N {0} where for p R N \ {0}, p = p. It turns out see for instance [3, 33] that the level set p Γ t = { x R N : u x, t = 0 } can be seen as a generalized evolution of the set Γ 0 with normal velocity 1.4 V = κ + c 1 x where κ is the mean curvature of the hypersurface Γ t where it is smooth, and the normal is by convention the outward normal to the set { x R N : u x, t > 0 }. When this set is convex, the mean curvarure is non positive. It is known from [5] that equation 1.3 admits Lipschitz correctors for Lipschitz Z N -periodic function c satisfying moreover the condition 1.5 inf y R N c y N 1 Dcy > 0. But the question was left open to know if 1.5 is necessary for homogenization or whether 1.6 inf y R N cy > 0 is enough, as it is the case when there is no curvature term in Main results It turns out that condition 1.6 is not enough to get homogenization in dimension N 3 as shows the following counter-example. We use the notation x = x 1,..., x N R N. Theorem 1.1 Counter-example to homogenization in dimension N 3 Let N 3. Then there exists a function c C R N which is Z N -periodic, satisfying 1.6 and moreover with cx independent on the variable x N, such that the following holds. For the initial data u 0 x = x N, the solution u of 1.3 satisfies for some constants c > c 1.7 lim sup u x, t ct x N > ct x N lim inf u x, t for all t > 0 x, t, x,t,0 x, t, x,t,0 i.e. there is no strong limit, and hence homogenization does not take place. 3

4 On the contrary in dimension N =, condition 1.6 is sufficient to get homogenization as we will see below see Theorem 1.4. Indeed in dimension N =, homogenization holds for general equation 1.1 with F safistying certain conditions. Let us define D 0 := S N R N \ {0} R N, where S N denotes the set of real symmetric N N matrices. We assume that F X, p, y has arguments X, p, y D 0 and satisfies the following properties: Assumption A A1 Degenerate ellipticity: F CD 0 and for all X, p, y D 0, we have F X + Q, p, y F X, p, y for all Q 0 with Q S N A F is geometric: for all X, p, y D 0, we have F λx + µp p, λp, y = λf X, p, y for all λ > 0, µ R A3 Z N -Periodicity: for all X, p, y D 0, we have F X, p, y + k = F X, p, y for all k Z N A4 Regularity: this technical assumption is given in Subsection 3.. We will also assume the following Assumption B: Bound from below: There exists δ > 0 such that for all arguments 0, p, y D 0, we have 1.8 F 0, p, y δ p. In order to keep simple the presentation, we chose not to give the details of the classical but technical regularity assumption A4 in this introduction. Under assumption A, a comparison principle holds see Theorem 3.3. Remark 1. Notice that assumptions A1, A3, and B imply that there exist constants C 0, c 0 > 0 and R > / such that for all p, y S N 1 R N, we have 1.9 C 0 F 0, p, y F 1R I, p, y c 0 > 0. Then we have the following result. Theorem 1.3 The cell problem in D Assume that N = and that A and B hold. Then for any p R N, there exists a unique 4

5 real number F p with F p > 0 if p 0 and F 0 = 0 such that there exists a bounded Z N -periodic function v : R N R solution of 1.10 F p = F D v, p + Dv, y on R N. We can choose v such that 1.11 sup v inf v κ 0 p with κ 0 := 100 R C 0 c 0 where R, C 0, c 0 are given in 1.9. Moreover the map p F p is continuous and positively 1-homogeneous, i.e. for any p R N F λp = λ F p for any λ 0. Let us mention that under assumptions A and B, in the case where F X, p, y 1, y is independent on y, the existence and uniqueness up to addition of constants of a corrector v when p R \Re 1, has been established in Lou [8] see also Lou, Chen [9] and Chen, Namah [14], for particular cases. As a consequence, we can show the following homogenization result with an Ansatz that looks like p x + t F p + v 1 x, but contrarily to the common belief, is much more involved than the classical perturbed test function method due to Evans. The main difficulty is created by the discontinuity of the Hamiltonian F when the gradient vanishes: Theorem 1.4 Homogenization of geometric motions in D Assume that N = and that A and B hold. Let us consider the solution u of 1.1 with initial data u 0 which is uniformly continuous on R N. Then u converges locally uniformly on compact sets of R N [0, + towards the unique solution u 0 of 1. with the function F given by Theorem 1.3. Indeed, Theorem 1.4 appears to be a corollary of a more general result in any dimension Theorem 1.5, for which we need to introduce the following assumption: Assumption B : Perturbed correctors: We set for η > 0: F η X, p, x = sup F X, p, y, y x η resp. F η X, p, x = inf y x η F X, p, y. For all p R N, there exists η 0 > 0 and κ 0 > 0 such that for all η [0, η 0, there exists a corresponding Z N -periodic function v η resp. v η and a real number F η η 0 p resp. F η η 0 p such that F η = F η D v η, p + Dv η, y resp. Fη = F η D v η, p + Dv η, y on R N such that for v = v η, v η, we have sup v inf v κ 0. Then we have: 5

6 Theorem 1.5 Conditional homogenization in dimension N when perturbed correctors do exist Assume that N and that A and B hold. Let us consider the solution u of 1.1 with initial data u 0 which is uniformly continuous on R N. Then u converges locally uniformly on compact sets of R N [0, + towards the unique solution u 0 of 1. with the function F = F 0 = F 0 given by assumption B. With an assumption weaker than B allowing negative normal velocities, namely assumption B in Section 10, it is still possible to get a corrector Theorem As an interesting application of Theorem 1.5, it is for instance possible to get homogenization results in D of equation 1.3 with certain sign changing velocities see Theorem Brief review of the literature The first results of uniqueness for the mean curvature motion, were obtained by Evans, Spruck [1] and Chen, Giga, Goto [13]. For general presentations of viscosity approaches to the motion of fronts, see Giga [], Souganidis [3, 33], Ambrosio [1], Soner [31]. One of the main difficulty with the evolution of fronts is the possibility of fattening see Barles, Soner, Souganidis [4]. The homogenization of Hamilton-Jacobi equations was pionered in Lions, Papanicolaou, Varadhan [6], and then extended to the fully non linear uniformly elliptic case in Evans [19, 0]. The case of geometric equations was studied only recently. In Lions, Souganidis [5], in any dimensions N 1, a Lipschitz bound on the correctors associated to forced MCM equation 1.3 is shown under assumption 1.5 and also for more general equations under suitable assumptions. In Cardaliaguet, Lions, Souganidis [6], it is in particular shown that in dimension N =, if cy = gy 1 with g > 0, and 0 g min g < [0,1] [0,1] then for any p R, there exists a Lipschitz continuous corrector v only depending on y 1 solution of Moreover F p > 0 if p R \Re 1, and F p = 0 if p Re 1. Among other things, in dimension N =, a counter-example to homogenization is also given in a case where g = 0 see also Remark 4.3. [0,1] In Cesaroni, Novaga [9], still in dimension N =, it is in particular shown that for p = e, there exists a Lipschitz continuous corrector v only depending on y 1 if or if [0,1] g > 0, and min [0,1] [0,1] g 0 and max [0,1] g > 0. g min g < 3 [0,1] More generally, it is shown in dimension N 1, that if cy = gy 1,..., y N 1 this is the case of a laminate, and if A T N 1, g > PerA, T N 1 A 6

7 then there exists a pseudo corrector v only dependending on y = y 1,..., y N 1 and an open set E T N 1 such that v is locally bounded on E and v = on T N 1 \E. The pseudo corrector is a kind of pseudo travelling wave. Notice that our counter-example Theorem 1.1 provides an example of a case where such a pseudo corrector is not a true corrector in the case g > 0 in dimension N 3. Under certain assumptions, the homogenization result of [9] has been extended in [10] to the case with an additional drift term given by a gradient vector field. Let us mention Craciun, Bhattacharya [15], where a formal assymptotics of F p is given in the limit λ + for a geometric motion given by V = λκ + c. On the other hand, it is shown in Dirr, Karali, Yip [18], that for a geometric motion V = κ + δc with c C T N without any sign condition on c, if δ > 0 is small enough, then for any p R N, there exists a Lipschitz continuous corrector v solution of 1.10, which is moreover unique if F p 0. Part of the method of proof is based on the arguments of Caffarelli, De La Llave [5] for the construction of minimal surfaces in a periodic setting. See also Chambolle, Thouroude [1] for a BV approach of the result in [5]. It is shown in particular in [1], that if 1.1 c = 0 and µ 0, 1, A T N, c µ PerA, T N T N then, for any p R N, there exists a corrector v and F p = 0. Let us mention that the homogenization of geometric motions V = κ + 1 c 1 x has been done in Barles, Cesaroni, Novaga [3] under the assumption that cy = gy and that 1.1 holds with c, T N replaced by g, T N 1. The case of a geometric motion in dimension N = V = κ + c 1 x with cy = gy 1, has been studied in Cesaroni, Novaga, Valdinoci [11]. 1.4 Organization of the paper The paper is organized as follows. In Section, we present the strategies of our main proofs. In Section 3, we recall basic properties of viscosity solutions. In Section 4, we do the proof of Theorem 1.1 about the counter-example to homogenization in dimension N 3. In Section 5, we present preliminary results on the evolution of the front, including the connectedness property Proposition 5.9 and the black ball barrier Proposition In Section 6, we prove the flatness of the front using Section 5. In Section 7, we prove Theorem 1.3, i.e. we show the existence of a corrector for the cell problem. In Section 8, we prove Theorem 1.5 about the conditional homogenization in any dimension. In Section 9, we prove Theorem 1.4 about the homogenization in D. In Section 10, we prove the existence of correctors in D Theorem 10.3 under a general assumption B which allows sign changing normal 7 A

8 velocities. In Section 11, we present some examples and applications, both for the forced MCM and the G-equation. Finally the appendix Section 1 contains three subsections, respectively about barriers, inf-convolutions and the proof of the comparison principle Theorem 3.3. We did not find precisely in the literature the result we need for the comparison principle, even if its proof is essentially based on [3]. We expect that this detailed proof will be of future use for other authors. Strategy of the proofs We discuss here the ideas underlying the proofs of the main results..1 The counter-example in 3D Theorem 1.1 The basic idea is that in dimension 3 or higher dimension we can find unbounded convex sets with negative curvature which are invariant by the geometric motion given by the normal velocity V = κ + c with c = 1. This is the case of cylinders whose section are circles. Then we can perturb the velocity c inside the cylinder and outside the cylinder in order to allow the propagation of fronts which are asymptotic to the cylinder with different velocities inside the cylinder and outside the cylinder. Considering periodic copies of the cylinder, we can construct a periodic velocity c which does not depend on the coordinate along the cylinder. Then the two fronts inside and outside the cylinders can be used as barriers to show that homogenization can not occur at least in a strong sense. Notice that the analogue in dimension of the cylinder is simply a circle, which does not allow the propagation of a front inside the disk!. The cell problem in D Theorem 1.3 The idea of the construction of a corrector is purely geometric even if we find convenient to use a level set formulation to work. Under assumption B, we can think that the front propagates as a fire. This means that the front never comes back. Therefore we can distinguish the burnt region black region and the unburnt region white region. Moreover if our initial black region is a half plane, then at least in some weak sense, we can show that the black region stays connected for all time. The basic phenomenon to avoid is the creation of a very thick transition region between the black and the white region like on Figure 1. A bounded connected component of the white region like in the bottom of Figure 1 can exist, but has to be thin enough. Indeed, it can not contain a unit square, otherwise by an integer translation argument the Birkhoff property, it will contain infinitely many such squares. Notice that all such bounded white components will diseapear, because they are contained in a white ball surrounded by the black region that will itself disappear in finite time depending on its size. This remark is not sufficient, because we need to bound the time after which they have disappeared. The situation is even worth if we have very long fingers like in Figure 1. We have to show that these fingers will disappear sufficiently quickly. 8

9 mean upper front at time t Figure 1: A typical situation to avoid: thick transition with long fingers The fundamental remark is that assumption B also implies the existence of a black ball of sufficiently large radius R > 0, which can increase or propagate in any direction. This black ball can be used as a barrier that will clean the white region remaining pinned in the black region. This black ball can be used to show that afer a fixed time T > 0, the new picture will be necessarily like on Figure, with a bounded thickness of the transition region between the black and white parts. mean upper front at time t+t mean upper front at time t bounded thickness cleaning the white parts up to time t+t Figure : Cleaning the picture after a fixed time T > 0 The cleaning phenomenon is possible because the boundary of the white long fingers is connected and then in D locally separates the plane in two big parts W for white and B for black, like locally two half planes if the white finger is straight enough. Then we can introduce see Figure 3 the black ball in the part B which is no longer true in higher dimensions, like it is shown in the counter-example with cylinders in dimension 3 for instance and propagate the black ball in the direction of the part W. This process cleans 9

10 the white part W and make disappear the white finger in a fixed finite time at least in the direction of the thickness of the finger. region W region B one boundary of a white finger the black ball motion of the black ball Figure 3: Cleaning the white part with the black ball barrier Once we are able to show that the thickness of the transition region between the white and black region is bounded uniformly in time, this shows that the front is roughly flat. This property is sufficient to show that the flat front propagates with a well-defined velocity. Passing to the limit as the time goes to infinity, it is then possible to define a corrector which describes the periodic propagation of the flat front in the periodic framework..3 Homogenization Theorem 1.5 The goal of this subsection is to give some heuristic explanations of the difficulties arising in the homogenization of geometric equations, and the main arguments that we have introduced in our proof of Theorem 1.5. Try 1: the naive approach and the difficulty when the gradient vanishes The naive try is the following perturbed test function for a corrector w.1 ϕ x, t = ϕx, t + wx/. It is a common belief see for instance [15] and [5] that once we are able to show the existence of correctors, then the homogenization result is a corollary obtained using Evans perturbed test function method see [19]. The point is that this belief is false when we want to homogenize equations like mean curvature motion, because the Hamiltonian is discontinuous when the gradient vanishes. More precisely, in the following, we recall the classical Evans method and then present the difficulty we have to face. 1.1 The classical Evans method If ϕ is a test function touching u := lim sup u from above and which does not satisfy the 0 subsolution viscosity inequality, i.e.. ϕ t > F Dϕ at some point P 0 10

11 then given a super corrector w associated to p = DϕP 0, we hope that the perturbed test function ϕ given by.1 satisfies.3 ϕ t F D ϕ, D ϕ, x/ in a neighborhood of the point P 0 in order to get later a contradiction with the fact that ϕp 0 = up The difficulty Inequality.3 means for y = x/ and P = x, t.4 ϕ t P F D ϕp + D wy, DϕP + Dwy, y with P close to P 0, i.e. for a point P in a neighborhood of P 0, which is independent on for small enough. Notice that. means for all y:.5 ϕ t P 0 > F DϕP 0 F D wy, DϕP 0 + Dwy, y but.5 does not imply in general.4 for small quantities ϕ t P ϕ t P 0, DϕP DϕP 0 and D ϕp. The difficulty comes from the fact that, even if F DϕP 0 > 0, it may happen that.6 y 0 such that DϕP 0 + Dwy 0 = 0 and.7 the curvature of the level set D wy DϕP 0 + Dwy blows up when y y 0. Then, it is not clear at least for us how to avoid the case where a small perturbation q = DϕP + Dwy 0 would satisfy q 0 with q in a direction such that F D wy 0, q, y 0 F D wy 0, 0, y 0 > ϕ t P 0 > F D wy 0, 0, y 0. Try : New ingredients Given a parameter η > 0, the new perturbed test function is the following.8 ϕ x, t = inf ϕ z, t. z B η x From Try 1, it is clear that we can not really hope to construct a perturbed test function like ϕ which is a supersolution in a neighborhood of a point P 0. We show here that, in order to get a contradiction, we only need this perturbed test function to be a strict supersolution at a one contact point with u, which is much easier to check. Of course, we also need to control the curvature of the level set at that contact point. In what follows we present some ideas to reach our goal..1 A pointwise Evans method The first main idea is to replace the standard Evans method, by the following Pointwise Evans method. Let us define for some general function u: F [, u] := F D u, Du, x/. We just consider a local maximum point P close to P 0 as goes to zero of u ϕ. We formally have at P u t = F [, u ] F [, ϕ ] < ϕ t 11

12 because we expect ϕ to be a strict supersolution. We then get a contradiction from the fact that u t = ϕ t at the point P. Notice that in order to have a strict supersolution, we still need to control the curvature of the interface and this is the goal of the next idea.. Geometric inf-convolution by balls We now introduce an argument of inf-convolution by balls, in order to bound the curvature from one side and then to avoid difficulty.7. We recall that, given a corrector w associated to a gradient p, the planar-like function solves ly, τ = λτ + p y + wy, with λ = F p > 0.9 l τ = F D l, Dl, y. We then define the inf-convolution by balls of radius η > 0: l η y, τ = inf lz, τ. z B ηy Then for any a R each upper level set {l η > a} has exterior tangent balls of radius η at each point of its boundary, which implies that its curvature matrix is bounded from above by 1 I see Figure 4. η l < 0 motion l = 0 η l > 0 η l = 0 η Figure 4: The new interface after inf-convolution by balls This implies in particular that D F D l η l η, Dl η, y = Dl η F Dl η, Dl η Dl η, y Dl η F 1 η I, Dl η Dl η, y c η Dl η for some constant c η > 0..3 Bound from below on the gradient x Notice that ϕ x, t looks like l, t, and then its natural to replace ϕ by ϕ given in.8, and to look at a local maximum point P close to P 0 of u ϕ. Therefore we have at P : 0 < 1 F DϕP 0 ϕ t = u t = F [, u ] F [, ϕ ] c η D ϕ 1

13 which shows that the gradient D ϕ is bounded from below by a constant depending only on η..4 Difficulty for checking that ϕ is a strict supersolution at P We have for P = x, t : ϕ P = ϕ P for some point P = x, t with x B η x. On the one hand, we get ϕ t P = ϕ t P = ϕ t P > F DϕP 0 for small enough. On the other hand, we have with ȳ = x, ỹ = x F [, ϕ ] P = F D ϕ P, D ϕ P, x = F D ϕ P, D ϕ P, ȳ = F D ϕ P + D wỹ, Dϕ P + Dwỹ, ȳ for which we have { D ϕ P = Dϕ P + Dwỹ is bounded from below, ȳ ỹ η. And in order to conclude that ϕ is a strict supersolution at P, it is enough to show that.10 F D ϕ P +D wỹ, Dϕ P +Dwỹ, ȳ F D wỹ, DϕP 0 +Dwỹ, ỹ = F DϕP 0. We consider here a small perturbation of the arguments of F. Because F is not uniformly continuous on the set where the gradients are bounded from below, we still need the following property:.11 D ϕ P, D ϕ P C which is not true in general. Try 3: Further regularization Given a parameter ρ > 0, the new perturbed test function is the following.1 ϕ x, t = inf ϕ x z 4 z, t + = ϕ x z 4 z, t + z R N 4 3 ρ 4 3 ρ z=z x. 3.1 Classical regularization In order to control the quantities in.11, this is natural to introduce the inf-convolution.1. Classically, this kind of inf-convolution is convenient for mean curvature type PDE, because the function 4 has zero second derivatives when its gradient is zero. Notice that here we could have taken another inf-convolution, because the case where the gradient 13

14 vanishes is already avoided by the bound from below on the gradient. For.1, we can show that for ρ small which implies z x x Oρ 1 4 D ϕ Oρ 1, Dϕ Oρ 1 4 which will give Difficulty The drawback of this regularization by inf-convolution is that in.10, it will make move the contact points ȳ into points y where now we have the estimate: y ỹ y ȳ + ȳ ỹ Oρ η. We still have to face the lack of uniform continuity of F as ρ goes to zero. Try 4: Our definitive choice We consider the test function ϕ given by.1 where the corrector w appearing in ϕ see.1 has to be replaced by w η associated to the Hamiltonian: F η X, p, x = We choose ρ small enough satisfying sup F X, p, y. y x η y ỹ Oρ η η and the adjustment of the parameter η is done such that the associated effective Hamiltonian F η is close enough to F = F 0 in order to satisfy ϕ t P 0 > F η DϕP 0. This last choice allows us to conclude the raisonning. The previous method is used to show that lim sup u is a subsolution of the limit equation. A similar but adapted method because we may have ϕ t P 0 < 0 is used to show that 0 lim inf u is a supersolution. 0 Remark.1 Link between 1.5 and inf-convolution by small balls It is possible to see that assumption 1.5 implies that if a characteristic function χx, t is a solution of 1.3 in dimension N with = 1 to fix the ideas, then χ η x, t = sup χy, t and χ η x, t = inf χy, t y B η x y B ηx are respectively sub and supersolutions for η > 0 small enough, when the total mean curvature of the smooth moving boundary satisfies κ Dc which is the case if κ c. 14

15 3 Properties of viscosity solutions 3.1 Viscosity solutions Let Ω R N be an open set and let T 0, + ]. We consider solutions u of the following equation 3.1 u t = F D u, Du, y on Ω 0, T =: Ω T with boundary - initial data 3. u = g on Ω {0} Ω [0, T =: p Ω T. For a general function u : Ω [0, T [, + ], we recall the definition of the upper resp. lower semi-continuous envelope u resp. u of u: u x, t = lim sup ut, s y,s x,t resp. u x, t = lim inf y,s x,t ut, s. We also recall that we say that u is upper resp. lower semi-continuous if and only if u = u resp. u = u. Given a function F continuous on D 0 = S N R N \ {0} R N, we also define for all X, p, x S N R N R N : { F X, p, x = lim 0 sup {F Y, q, y, Y, q, y D 0, X Y, p q, x y } F X, p, x = lim 0 inf {F Y, q, y, Y, q, y D 0, X Y, p q, x y }. Because of the continuity of F X, p, x for p 0, we have in particular F X, p, x = F X, p, x = F X, p, x if p 0. We are now ready to recall the definition of a viscosity solution Definition 3.1 Viscosity solution We use the previous notation. i Sub/super/solution of 3.1 We say that u : Ω [0, T [, + ] is a subsolution resp. a supersolution of 3.1 if u < + resp. u > and u is upper resp. lower semi-continuous and if for any P 0 = x 0, t 0 Ω T, if there exists some r 0 > 0 such that B r0 P 0 Ω T ϕ C B r0 P 0 such that { u ϕ on Br0 P 0 u = ϕ at P 0 then we have at P 0 ϕt F D ϕ, Dϕ, x0 resp. { u ϕ on Br0 P 0 u = ϕ at P 0 resp. ϕt F D ϕ, Dϕ, x 0. and a function We say that u is a viscosity solution of 3.1 if u is a subsolution and if u is a supersolution. ii Sub/super/solution of A function u : Ω [0, T [, + ] is said to be a subsolution resp. a supersolution of if it is a subsolution resp. supersolution of 3.1 and if furthermore it satisfies u g on p Ω T resp. u g on p Ω T. We say that u is a viscosity solution of if u is a subsolution and u is a supersolution. 15

16 3. The technical assumption A4 We give below the precise assumption A4. A4 Regularity: i Boundedness close to p = 0: For all R > 0, there exists a constant C R > 0 such that for all F X, p, y C R for all X R, 0 < p R y R N ii Variations in X, x: There exists K 9 and σ K : [0, + [0, + satisfying σ K 0 + = 0, such that we have F X, αx y, x F Y, αx y, y σ K { x y 1 + α x y } for all α 0 and X, Y S N, x, y R N satisfying I 0 X 0 I I Kα Kα 0 I 0 Y I I with α = 0 if x = y This kind of regularity assumptions are given partially page 443 in [4]. Remark 3. Notice that condition A4ii joint to the geometric property of F assumption A imply 3.3 F 0, 0, y = 0 = F 0, 0, y. Notice that we also have 3.4 F X, p, y = F Πp X Πp, p, x with Πp = I ˆp ˆp if p 0 which follows from assumptions A1-A and Theorem page 48 of []. Then we have the following result. Theorem 3.3 Comparison principle Assume that either Ω = R N or that Ω is a bounded open set of R N, and assume A. If u is a subsolution of 3.1 and v is a supersolution of 3.1 such that 3.5 lim sup { ux, 0 vy, 0, x, y R N, x y θ } 0 if Ω = R N θ 0 u v on p Ω T if Ω is a bounded open set then u v on Ω T. 16

17 3.3 An example Let us consider for instance the following natural example 3.6 F X, p, y = tr { Σ T p, y Σ p, y X } { Hp, y positively 1-homogeneous in p +Hp, y with Σ p, y I p p = Σ p, y where Hp, y and Σ p, y are Z N -periodic in y. We assume the following regularity H CR N R N ; R and Σ CS N 1 R N ; R N N, and that there exists a constant L > 0 such that 3.7 Hp, x Hp, y L x y p and Σ p, x Σ p, y L x y. Notice that equation 1.3 corresponds to the particular subcase and assumption B means Σp, y = I p p and Hp, y = cy p cy δ > 0. More generally, if we assume moreover that H satisfies the bound from below 1.8, then F also satisfies B. Checking A for F given by 3.6. We claim that F given by 3.6 satisfies A. The only thing non trivial to check is A4ii in the special case where H 0. We consider X, Y satisfying with α Kα I 0 0 I X 0 0 Y I I Kα I I Then for p = αx y 0, we multiply on the left by Σ p, x, Σ p, y and on the right by Σ p, x, Σ p, y T, and we get tr { Σ T p, xσ p, xx Σ T p, yσ p, yy }. 3.9 Kα Σ p, x Σ p, y KL α x y where we have used 3.7 in the second line. If x = y, then α = 0 in 3.8 implies X = Y = 0 and then 3.9 still holds. This shows A4ii with σ K a = KL a. 3.4 More properties on viscosity solutions The following results are classical in the theory of viscosity solutions see [17]. Proposition 3.4 Stability i semi-limits If u is a sequence of subsolutions resp. supersolutions of 3.1, let ux, t = lim sup x, t, x,t,0 u x, t, ux, t = lim inf x, t, x,t,0 17 u x, t.

18 If u < + resp. u >, then u is a subsolution resp. u is a supersolution of 3.1. ii suppremum/infimum Let S be a set of functions w such that w is a subsolution resp. a set of functions w such that w is a supersolution of 3.1, and u = sup w w S resp. u = inf w. w S If u < + resp. u >, then u is a subsolution resp. u is a supersolution of 3.1. Proposition 3.5 Perron s method Let u + resp. u be a supersolution resp. subsolution of 3.1 satisfying u u +. Then there exists a viscosity solution u of 3.1 satisfying u u u +. Proof of Proposition 3.5 The proof is essentially based on [13] and [4]. We repeat it for completness. We call S = { w : such that w is subsolution of 3.1, w u +} u. We define ux, t = sup w S wx, t. From the stability result Proposition 3.4 ii, we know that u is a subsolution. Assume that u is not a supersolution and let us get a contradiction. Then there exists a point P 0 = x 0, t 0 and a test function ϕ C B r0 P 0 for some r 0 small enough such that B r0 P 0 Ω T and such that u ϕ on B r0 P 0 and u = ϕ on P ϕ t = θ + F D ϕ, Dϕ, x 0, with θ > 0. Up to replace ϕp by ϕp P P 0 4, we can assume that 3.11 u ϕp P P 0 4. Notice that u P 0 < u + P 0 because otherwise ϕ would be a test function for u + and 3.10 would be in contradiction with the fact that u + is a supersolution. Therefore there exists some small δ 0, r 0 / such that ϕ t F D ϕ, Dϕ, x 3.1 for P B δ P 0. ϕp + δ 4 / u + P From 3.11, we deduce that 3.13 up u P δ 4 / ϕp + δ 4 / for P B δ P 0 \B δ P 0. 18

19 We now define wp = Notice that from 3.13, we have { maxϕp + δ 4 /, u P, P B δ P 0 u P, P Ω [0, T \B δ P 0. wp = maxϕp + δ 4 /, u P for P B δ P 0 and then w is a subsolution as the maximum of two subsolutions on B δ P 0 see Proposition 3.4 ii. This implies that w is a subsolution everywhere and from 3.1 that w S. On the other hand, we have 0 = u ϕp 0 = lim η 0 inf {u ϕp, P P 0 η}. Therefore there exists some P 1 B δ P 0 such that u ϕp 1 < δ 4 /, which implies that up 1 < wp 1. This is in contradiction with the definition of u as the maximal subsolution. This ends the proof of the Proposition. 3.5 Subdifferentials For later use, we recall here the definitions of sub/superdifferentials. Definition 3.6 Sub/superdifferentials Let x, t ux, t be a upper semicontinuous resp. lower semi-continuous function defined on an open set. For P 0 = x 0, t 0, we say that τ, p, X P,+ up 0 resp. τ, p, X P, up 0 if there exists a C test function ϕ such that u ϕ resp. u ϕ with equality at P 0 and τ, p, X = ϕ t, Dϕ, D ϕ at P 0. Remark 3.7 If ux, t is independent on t, we say that p, X D,± if an only if 0, p, X P,±. Definition 3.8 Limit sub/superdifferentials Let x, t ux, t be a upper semicontinuous resp. lower semi-continuous function defined on an open set. For P 0 = x 0, t 0, we say that τ, p, X P,+ up 0 resp. τ, p, X P, up 0 if there exists exists sequences such that τ k, p k, X k P,+ up k resp. τk, p k, X k P, up k such that τ k, p k, X k, up k τ, p, X, up 0. 19

20 4 Counter-example in dimension N 3 We set x = x 1,..., x N 1 such that x = x, x N. We now consider solutions U of the first line of 1.3 in the case = 1 for a velocity cx replaced by some general velocity c x which is independent on x N, i.e. U solution of { U t = tr D U I DU DU } + c x DU. We look for particular solutions Ux, t = ux, t x N which means that u solves at least for smooth solutions u u t 4.1 = c x Du + div. 1 + Du 1 + Du Then we have Proposition 4.1 Traveling profiles with different velocities for N 3 Let N 3. There exists a radial function c C R N 1 which is positive, and radial functions u + : R N 1 [, +, u : R N 1, + ] and constants c +, c satisfying u + < u and c + > c such that the profiles are solutions of 4.1. We have c ± t + u ± x 4. u + x = { u0 x if x < 1 if x 1 and u x = { + if x 1 u 0 x if x > 1 with u 0 C B 1 and u 0 C R N 1 \B 1. Moreover there exists r 0 > 1 such that { c x = c for x r 0 u x = constant for x r 0. The profiles of Proposition 4.1 are illutrated on Figure 5. Proof of Theorem 1.1 Using Proposition 4.1, we first define for R > r 0 a velocity defined on the centered square Similarly we define c R x = c x for x [ R/, R/] N 1. U R ± x, t = c ± t + u ± x x N for x [ R/, R/] N 1. Moreover, up to add a suitable constant to u + resp. u, we can assume that 4.3 u + 0 u. 0

21 c + u + u _ u _ c _ c _ x x =1 Figure 5: Profiles u ± with velocities c + > c We then extend by periodicity c R x and U± R x, x N, t as RZ N 1 -periodic functions for x R N 1. We get that U± R are both solutions of { U t = tr D U I DU DU } + c R x DU. Then the new functions Ū ±x, t := R 1 U R ± R 1 x, R 1 t are solutions of 1.3 with the velocity cx = Rc R Rx which is a positive smooth Z N -periodic function independent on x N. Using 4.3, we see that we have 4.4 Ū +x, t u x, t Ū x, t at time t = 0. From the comparison principle, we deduce that 4.4 holds true for all time t 0. Setting c = Rc +, c = Rc we deduce 1.7 from the fact that for t > 0 lim sup Ū+ x, t ct x N > ct x N lim inf Ū x, t. x, t, x,t,0 x, t, x,t,0 This ends the proof of the theorem. Proof of Proposition 4.1 Step 1: preliminary For the radial functions u 0 and c, we make the abuse of notation u 0 x = u 0 r and c x = c r for r = x with x R n and n = N 1. 1

22 We define the function ζ by the relation 4.5 u 0 = ζr x x. Then we easily see that a function c t + u 0 r is solution of 4.1 if and only if { } c 4.6 c r = κ with κ = ζ + n 1 ζ. 1 + ζ 1 + ζ r 1 + ζ We look for a function ζ which blows up at r = 1 and is smooth for r 1 such that we can take { c+ if r [0, c = c if r > 1 and we want to check that c given by 4.6 is nevertheless smooth and positive. Step : first computation As a first candidate for ζ, we propose 4.8 ζr = e 1 1 r 1 1 for r [0,. After some computations, we get 4.9 c r = c e 1 1 r 1 + sign1 r e and 1 e 1 1 r 1 r 1 r 1 e 1 1 r 1 +n r 1 1 e = r 3 r4 + Or 6 which is then a smooth function up to r = 0 analytic close to r = 0. With the choice 4.7 for any constants c ±, this shows that c is C for r <. Step 3: conclusion In order to define a function c r for all r, we simply set ζ = ζψ where ψ C [0, + is a cut-off function satisfying { 1 for 0 r 1 + η ψr = 0 for r 1 + η r 1 1 r 1 where η > 0 is small enough such that 1 + η <. We conclude choosing c + > c > 0 large enough such that c is positive. We finally get the profiles u 0 integrating 4.7 which provides the profiles u ± given by 4.. Notice in particular that ζ is not integrable in any positive or negative neighborhood of r = 1. This implies that lim u 0r = and lim sup u 0 r = +. r 1 This implies that c ± t+u ± x, even if they are unbounded, are solutions of 4.1 in the sense of Definition 3.1. This ends the proof of the proposition. r 1 +

23 Remark 4. In our example, we can deduce from 4.8 that 1 r 1 ln u 0 r as r 1. Notice also that in 4.6, we have c 1 = κ = n 1 which corresponds to the negative mean curvature of the tube of equation r = 1. On the contrary in dimension N = = n + 1, this curvature vanishes and then the velocity c too. Remark 4.3 Example of non homogenization in D with sign changing velocity In the case N =, i.e. n = 1 in 4.9, we can take any c + > 0 and c < 0 and the construction of Proposition 4.1 and Theorem 1.1 provides a non homogenization result for a sign changing velocity c x 1 which is R-periodic. 5 Preliminaries in any dimension We consider a solution ux, t of 5.1 u t = F D u, Du, y on R N 0, + with initial data 5. ux, 0 = u 0 x = x ν for x R N. Proposition 5.1 Existence and properties of the solution Assume A. Let ν S N 1 and u 0 x = ν x. Then there exists a unique solution u of Moreover u is continuous and ux, t ν x is Z N -periodic in x, and there exists a constant C > 0 such that u t C on R N [0, +. For any 0 < T < +, there exists a modulus of continuity m T such that 5.3 ux, t uy, t m T x y for all x, y R N, t [0, T ]. If we assume moreover B, then we have 5.4 u t δ > 0 on R N [0, +. Proof of Proposition 5.1 Step 1: barriers, existence, uniqueness We set the barriers sub/supersolutions u ± x, t = u 0 x + C ± t with ± C ± = sup x R N ±F 0, ν, x. 3

24 Then by Perron s method Proposition 3.5, there exists a solution u of 5.1 satisfying which implies in particular u u u + u, 0 = u 0. Then u solves Furthermore we deduce from the comparison principle Theorem 3.3 that this solution is unique and is then continuous. Step : periodicity For any k Z N, we have u 0 x + k = u 0 x + ν k. The comparison principle implies that ux + k, t = ux, t + ν k i.e. ux, t ν x is Z N -periodic in the x variable. Step 3: time regularity Let h 0. Then we have 5.5 ux, t + h ux, t + C + h for t = 0 and the comparison principle implies that 5.5 holds for every time. Similarly, we get that 5.6 ux, t + C h ux, t + h. Then 5.5 and 5.6 show that C u t C +. Notice that this result joint to the continuity of u and to the periodicity of ux, t ν x implies the existence of a modulus of continuity as in 5.3. Step 4: further result under assumption B Then we have C δ and this implies 5.4. This ends the proof of the proposition. Proposition 5. No fattening Assume A and B and let u be the function given in Proposition 5.1. i No fattening Then u satisfies for all t 0: 5.7 Int { x R N, ux, t = 0 } =. As a consequence, the sets E t = { x R N, ux, t 0 } and E o t = { x R N, ux, t > 0 } only differ on a set of empty interior and E o t, E t { x R N, ux, t = 0 }. 4

25 ii monotonicity We have 5.8 E t E s resp. E o t E o s for all s t 0. iii stability of E We have 5.9 E s = E t s>t and Es o = Et o. s<t Remark 5.3 Notice that Et o Int E t, but we may have Int E t Et o if for instance u, t is positive on B 1 0\ {0} and vanishes at x = 0. Similarly, we have Et o E t, but we may have E t Et o if for instance u, t is negative on B 1 0\ {0} and vanishes at x = 0. Remark 5.4 We can even show that 5.10 Es o = E t s>t and E s = Et o. s<t Proof of Proposition 5. Proof of i Assume that there exists t 0 > 0 such that there exists x 0 and r > 0 such that 5.11 B r x 0 { x R N, ux, t 0 = 0 }. Given such r > 0 and some 0, 1, we consider the test function φ x, t = A r x x Ā t t 0 for x B r x 0, t t 0 where A r > 0 and Ā are constants that we will fix later. If x B r x 0 and t [0, t 0 + 1] recall that ux, t ux 0, t m t0 +1r where the modulus of continuity m t0 +1 is given in 5.3. Moreover, for t t 0, we have ux, t ux, t 0 C. Therefore for A r := r 4 m t0 +1r > r 4 m t0 +1r, and we have Ā := C > C Q = B r x 0 [t 0, t 0 + ] sup Q u φ > sup Q u φ. In particular there exists x, t Int Q such that sup Q u φ = u φ x, t 5

26 and then 5.1 δ φ t F D φ, Dφ, x at x, t where we have used that u t δ see 5.4. We pass to the limit x, t x 0, t 0 as 0 with x 0 B r x 0. In particular we get that sup Q 0 u φ 0 = u φ 0 x 0, t 0 with φ 0 x, t = A r x x 0 4. Therefore 5.11 implies that x 0 = x 0. Passing also to the limit in 5.1, we get 0 < δ F 0, 0, x 0 = 0 where we have used 3.3 to identify to zero the right hand side. Contradiction. This implies 5.7. Proof of ii The monotonicity of u see 5.4 implies 5.8. Proof of iii The continuity of u implies 5.9. This ends the proof of the proposition. Proposition 5.5 Birkhoff property Using the notation of Proposition 5., let us define the set Then Moreover, if k A, then for all t 0 A = { k Z N, k + E 0 E 0 }. A = { k Z N, k E 0 } k + E t E t. Proof of Proposition 5.5 We simply notice that and k A if and only if We also notice that E 0 = { x R N, ν x 0 } ν k 0. u 0 x + k u 0 x. Therefore from the comparison principle and the invariance of the equation by integer translations, we deduce that ux + k, t ux, t. This implies 5.13 and ends the proof of the proposition. 6

27 Proposition 5.6 Characteristic functions Assume A and B. Let us consider the sets E t and Et 0 the following two functions defined in Proposition 5.1. Then are solutions of 5.1 and χ E, t := χ Et and χ E o, t := χ E o t for all t χ E = χ E o and χ E o = χ E. Proof of Proposition 5.6 Step 1: Proof of χ E o = χ E Let us consider a point P 0 = x 0, t 0. If up 0 0, then by continuity of u, we conclude that χ E o P 0 = χ E op 0 = χ E P 0. Now if up 0 = 0, then ux 0, t 0 + h δh for all h > 0 and then P h = x 0, t 0 + h E t0 +h. This implies that χ E o P 0 lim sup χ E op h = 1 h 0 and then χ E o P 0 = 1 = χ E P 0. Step : Proof of χ E = χ E o Similarly, if a point P 0 = x 0, t 0 is such that up 0 0, then by continuity of u, we get that χ E P 0 = χ E op 0. Because u t δ, we deduce that if up 0 = 0, then for all h 0 such that t 0 h 0, we have ux 0, t 0 ux 0, t 0 h δh, and then P h = x 0, t 0 h is such that up h < 0 for all h > 0. Therefore if t 0 > 0, which means χ E P 0 lim inf h 0 χ E P h = χ E P 0 = 0 = χ E op 0. If up 0 = 0 with t 0 = 0, we simply check that χ E0 = χ E o 0, which again implies Step 3: Sub/supersolutions We use an idea of [4]. Let us define for > 0 u x, t = β ux, t with β a = 1 { a } 1 + tanh. Notice that β is smooth and then, using the fact that the equation is geometric assumption A, it is easy to check that u is also solution of 5.1. Let us define u := lim sup u and u := lim inf u

28 Then we have using the pointwise limit of u as goes to zero χ E o u u χ E. Because by construction, u is lower semicontinuous and u is upper semicontinuous, we deduce from 5.14 that u = χ E and u = χ E o. By stability of viscosity solutions see Proposition 3.4 i, we deduce that u = χ E is a subsolution and u = χ E o is a supersolution. This ends the proof of the Proposition. Proposition 5.7 Bound from inside on the expansion of E t Let u be the solution given in Proposition 5.1. If ux 0, t 0 0, then for each τ > 0 ux, t 0 + τ 0 for x B r x 0 with r such that m t0 +τr δτ. In particular this implies that x 0 E t0 B r x 0 E t0 +τ. Proof of Proposition 5.7 Let α = ux 0, t 0 0. We have for τ 0 ux 0, t 0 + τ α + δτ and for t t 0 ux, t ux 0, t m t x x 0. Therefore for τ 0, we get This implies the result. ux, t 0 + τ α + δτ m t0 +τ x x 0. Corollary 5.8 Bound from outside for the forward evolution of E t Let u be the solution given in Proposition 5.1. If x 0 {u, t 0 < 0}, then for τ > 0 such that t 0 τ 0, we have ux, t 0 τ < 0 for x B ρ x 0 with ρ such that m t0 ρ δτ. In particular this implies that 5.16 E t0 τ E t0 \ B ρ x 0. x 0 E t0 8

29 Proof of Corollary 5.8 Just consider a sequence x n x 0 with x n {u, t 0 < 0} and apply Proposition 5.7 assuming by contradiciton that ux, t 0 τ 0 with x B ρ x n, in order to get a contradiction. This means which implies in particular x 0 R N \E t0 B ρ x 0 R N \E t0 τ Proposition 5.9 Arbitrarily long arc-connected components of the set Int E t Let x 0 Int E t0 with t 0 0 and ω 0 be the arc-connected component of Int E t0 containing x 0. Then for any r > 0, we have 5.17 ω 0 B r x 0. Proof of Proposition 5.9 Notice that ω 0 is an open set because Int E t0 is open. Assume that for some r > 0, 5.17 does not hold. Then this means that In particular, we also have ω 0 B r x 0. ω 0 Int E 0 = and t 0 > 0 using the fact that Int E 0 is arc-connected and unbounded. Let us define From 5.9, we deduce that t = inf {s [0, t 0 ], E s ω 0 } E t ω 0 = Case 1: t < t 0 Notice that s>t E s ω 0. ω 0 E t0 indeed if ω 0 Int E 0, then we get a contradiction with the definition of the arcconnected component ω 0. Because of Corollary 5.8, we deduce that for τ > 0 with t 0 τ 0 E t0 τ ω 0 ω 0 \ B ρ x 1 with m t0 ρ δτ x 1 ω 0 and then E s ω 0 ω 0 if s < t 0. Therefore, for s < t 0, the set E s ω 0 is a closed set and from 5.18 and the monotonicity of the sets, we deduce that E t ω 0 = Et +1/n ω 0. n N\{0} 9

30 There exists x n E t +1/n ω 0, from which we can extract a convergent subsequence with x n x ω 0. This shows that x E t +1/k ω 0 for all k > 0. Then Let us consider the function x E t ω 0. φx, t = t t + 1 which is a test function from above for χ E on ω 0 t, t +. We get at x, t : 1 = φ t F 0, 0, x = 0. Contradiction. Case : t = t 0 We get the same contradiction at any point x, t 0 with x ω 0. Proposition 5.10 The self-propagating ball barrier Assume that 1.9 holds and let us consider some ξ S N 1 and z 0 R N. For t 0, let us define the function χ G, t = χ Gt with G t = B R z 0 + c 0 sξ. Then χ G is a subsolution of 5.1 on R N 0, +. 0 s t Proof of Proposition 5.10 Let us consider a test function ϕ satisfying for some r 0 > 0 χ G φ on B r0 P 0 with equality at some point P 0 = x 0, t 0 R N 0, +. We want to check the viscosity inequality satisfied by ϕ. In particular, there exists a unique s 0 [0, t 0 ] such that x 0 B R z 0 + c 0 s 0 ξ and then Step 1: time derivative For τ R small, let us define Then we have for τ small enough This implies x 0 = z 0 + c 0 s 0 ξ Rp 0 for some p 0 S N 1. x τ = z 0 + c 0 s 0 + τξ Rp 0. ϕx τ, t 0 + τ χ G x τ, t 0 + τ = 1 with equality for τ = t ϕ + c 0 ξ Dϕ = 0 at P 0. 30

31 Step : gradient estimate Let us set y 0 = z 0 + c 0 s 0 ξ. We have 5.0 φ, t 0 1 on B R y 0 with equality at some point x 0 = y 0 Rp 0. This implies that 5.1 Dϕx 0, t 0 = p 0 Dϕx 0, t 0. Step 3: curvature estimate From 5.0, we also deduce that there exists a C function β satisfying β0 = 0 and 5. β 0 = Dϕx 0, t 0 when β 0 > 0 it is enough to take any β 0 < D ϕx, t 0 p, p for all x = y 0 Rp with p S N 1 close to p 0 such that This implies that ϕx, t 0 β x y 0 R in a neighborhood of x D ϕx 0, t 0 β 0 R I + β 0 p 0 p 0. Step 4: conclusion We get ϕ t = c 0 ξ Dϕ = c 0 ξ p 0 Dϕ c 0 Dϕ Dϕ F 1 R I, p 0, x 0 F D ϕ, Dϕ, x 0 where we have used 5.19 in the first line, 5.1 in the second line, 1.9 in the fourth line, and 5.3, A1, A in the last line. This shows that χ G is a subsolution and ends the proof of the Proposition. Remark 5.11 Notice that the function ux, t = R + c 0 t x is a subsolution in R N \B R 0 0, + and this can also be used to check that Proposition 5.10 holds true. 31

32 6 Flatness of E t for N = In order to simplify the description, we will use the analogy with the propagation of a fire. We call E t the burnt or black region and its complement R N \E t is called the unburnt or white region. Proposition 6.1 Black cubes Let us assume A and B and consider t 0 0. If x 0 Int E t0, then with τ = 5R/c 0 and R = / + R. x 0 + { x R, ν x R } E t0 +τ Proof of Proposition 6.1 Step 1: choice of a ball Let ω 0 be the connected component of Int E t0 containing x 0. From Proposition 5.9, we know that there exists a point y 0 B 4R x 0 ω 0 and a continuous path γ : [0, 1] B 4R x 0 ω 0 with γ0, 1] B 4R x 0, such that γ1 = x 0, γ0 = y 0. We set ξ := y 0 x 0 y 0 x 0. Let us call t the smallest t such that Then γ[0, t ] splits the half disk γt x 0 y 0 x 0 = 0. D + := {x B 4R x 0, x x 0 y 0 x 0 > 0} in two open connected components ω σ for σ = +, with ω ± { x D + B 4R x 0, ±x x 0 ξ > 0 }. See Figure 6. We also define the strip S x0,y 0 = { x R, 0 < x x 0 y 0 x 0 < y 0 x 0 = 4R } and the extensions of the sets ω ± as ˆω ± = ω ± {x S x0,y 0 \B 4R x 0, ±x x 0 ξ > 0}. The sets ˆω ± are also two connected open sets and we have the partition of the strip: S x0,y 0 = ˆω + ˆω γ[0, t ] S x0,y 0. Step : Using a self-propagating ball barrier Let z 0 = y 0 + x 0 /. For t t 0, the characteristic function of the set 0 τ t t 0 B R z 0 + ξ 5R + c 0 τ 3

33 ξ ω + x y 0 γ 0 B 4R x 0 ω_ Figure 6: Construction of ω + and ω ξ ω + z 0 x 0 y 0 B 4R x 0 ω _ z 5R ξ B R 0 Figure 7: The ball barrier propagating on ω + is a subsolution on ˆω + [t 0, + see Proposition 5.10 and Figure 7. Similarly, the characteristic function of the set 0 τ t t 0 B R z 0 + ξ5r c 0 τ is a subsolution on ˆω [t 0, +, and we deduce that for τ = 5R/c 0, we have { } y 0 x 0 E t0 +τ x B 4R x 0, R x z 0 y 0 x 0 R B R z 0 B / z 0 where we have used the fact that R /. Step 3: Using Birkhoff property From the Birkhoff property Proposition 5.5, we deduce that for any k Z such that k E 0, we have B / z 0 + k k + E t0 +τ E t0 +τ. 33